Mathematics for Business Administration: Multivariable Optimization PDF
Preview Mathematics for Business Administration: Multivariable Optimization
Mathematics for Business Administration: Multivariable Optimization Universidad de Murcia Mar´ıa Pilar Mart´ınez-Garc´ıa UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization These slides are intended for students of Business administration whose mathematical requirements go beyond the calculus for functions of one variable. The material includes a basic course on multivariable optimization problems, with and without constraints, and the tools of linear algebra needed for solving them. UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization Contents Chapter One: Convex sets. Convex and concave functions Chapter Two: Introduction to multivariable optimization Chapter Three: Classical optimization Chapter Four: Constrained Optimization Appendix: Matrix Algebra and Quadratic Forms Answers to the Problems Answers to the Multiple choice questions UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ReviewproblemsforChapter1 Multiplechoicequestions Chapter One: Convex Sets. Convex and Concave Functions ReturntoContents ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ReviewproblemsforChapter1 Multiplechoicequestions Outline Convex Sets Convex and Concave Functions ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ConvexSets ReviewproblemsforChapter1 Concaveandconvexfunctions Multiplechoicequestions Definition 1 Let x = (x ,x ,...,x ) and y = (y ,y ,...,y ) be any two points 1 2 n 1 2 n in Rn. The closed line segment between x and y is the set [x,y] = {z / there exists λ ∈ [0,1] such that z = λx+(1−λ)y} Definition 2 A set S in Rn is called convex if [x,y] ⊆ S for all x, y in S, or equivalently, if λx+(1−λ)y ∈ S for all x, y in S and all λ ∈ [0,1] Note in particular that the empty set and also any set consisting of one single point are convex. ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ConvexSets ReviewproblemsforChapter1 Concaveandconvexfunctions Multiplechoicequestions Intuitively speaking, a convex set must be ”connected” without any ”holes” and its boundary must not ”bend inwards” at any point. Convex Not convex ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ConvexSets ReviewproblemsforChapter1 Concaveandconvexfunctions Multiplechoicequestions Definition 3 A hyperplane in Rn is the set H of all points x = (x ,x ,...,x ) in 1 2 n Rn that satisfy p x +p x +···+p x = m 1 1 2 2 n n where p = (p ,p ,...,p ) (cid:54)= 0. 1 2 n Proposition 4 A hyperplane in Rn is a convex set. ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ConvexSets ReviewproblemsforChapter1 Concaveandconvexfunctions Multiplechoicequestions Definition 5 A hyperplane H devides Rn into two sets, H = {(x ,x ,...,x ) ∈ Rn/ p x +p x +···+p x ≥ m}, + 1 2 n 1 1 2 2 n n H = {(x ,x ,...,x ) ∈ Rn/ p x +p x +···+p x ≤ m}, − 1 2 n 1 1 2 2 n n which are called half spaces. Proposition 6 H and H are convex sets. + − ChapterOne:ConvexSets.ConvexandConcaveFunctions ChapterOne:ConvexSets.ConvexandConcaveFunctions Usefullinks ConvexSets ReviewproblemsforChapter1 Concaveandconvexfunctions Multiplechoicequestions Proposition 7 If S and T are two convex sets in Rn, then their intersection S ∩T is also convex. The union of convex sets is usually not convex. ChapterOne:ConvexSets.ConvexandConcaveFunctions
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